LOGIC AND SEQUENCES

“PRACTICE” FOR THE QUIZ7.5

LOGICproblem # 1

7+4+x+y+5 =32 and 7+4+x+z+5 =52 find the value for z-y.

SOLUTION #1

In the first problem, 16 + x + y = 32 or x + y = 16.

In the second, 16 + x + z = 52 or x+z = 36.

Solve for y: 16-x and solve for z: 36-x.

Then z-y = 36-x –(16-x) = 20

LOGICproblem # 2

Suppose 0.XY + 0.YX 0.XX X AND Y ARE DIGITS. What must Y

be?

SOLUTION #2

Think: I am adding Y to X but getting X back. It is the additive identity…which is ??? 0

SEQUENCES problem #3

32, 16, 8, 4 … What is the 8th term?

SOLUTION #3

Look for a pattern! Each number is the result of the preceding # being divided by 2, called a geometric sequence. Continue to get 2,1,1/2, ¼. The answer is ¼.

LOGICproblem #4

Suppose an “m-triple” is defined as (m/2 , m , 2m) for some number m.

Find an “m- triple” for the m value 60.

SOLUTION #4

Just substitute to find the triple. 60/2 and 2(60)

The triple would be: (30, 60, 120)

LOGICproblem #5

Let k # m be defined as the sum of all integers between k and m. For example: 6 # 9 = 7 + 8 = 15.

Find the value of ( 20 # 30) - ( 21 # 29)!

SOLUTION #5

Make a list: (21 + 22 +23 ….+29) – ( 22 +23 +…+28). Notice that all will “subtract out”

except for the 21 and the 29, so 21+29 = 50

SEQUENCESproblem #6

If a sequence has its nth term as 2n+3, find the sum of its third and sixth terms.

SOLUTION #6

The sequence is found by substituting in 1, then 2, then 3, etc to find the terms. So: 5,7,9,11,13,15. Now add terms 3 and 6: 9+ 15 =24

SEQUENCESproblem #7

Find the 10th term in the sequence: 1,2,3,5,8,13…

SOLUTION #7

Close inspection yields that a term is found by adding 2 previous terms: 1+2 =3; 2+3=5, etc so the 10th term is found after generating: 21, 34, 55, 89. The 10th term is 89.

LOGICproblem #8

If a is a nonnegative number and a* is defined as a* = sqrt(a) /3. When a* = 2, what must a be?

SOLUTION #8

Working backwards: sqrt(a) /3 = 2 says sqrt(a) would be 6. Therefore a = 36!

LOGICproblem #9

If r▲ t = rt – t for all integers r and t, find 6 ▲ 4.

SOLUTION #9

6 ▲ 4 = 6(4) -4 24 – 4 20

LOGICproblem #10

For x> 0, ♣ x is defined as 2x -1. What is the value of ♣6 / ♣3?

SOLUTION # 10

♣6 = 2(6) -1 =11 ♣3 = 2(3) -1 =5 11/5